Firooz Arash (a,b) , which have caused confusion in the normalization of ZEUS data is discussed and resolved. PCAC I. INTRODUCTION

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1 PION STRUCTURE FUNCTION F π 2 In te Valon Model Firooz Aras (a,b) (a) Center for Teoretical Pysics and Matematics, AEOI P.O.Box , Teran, Iran (b) Pysics Department, AmirKabir University (Tafres Campus), Hafez Avenue, Teran, Iran (January 3, 2003) Partonic structure of constituent quark (orvalon) in te Next-to-LeadingOrder is used to calculate pion structure function. Tis is a furter demonstration of te finding tat te constituent quark structure is universal, and once it is calculated, te Structure of any adron can be predicted tereafter, using a convolution metod, witout introducing any new free parameter. Te results are compared wit te pion structure function from ZEUS Coll. Leading Neutron Production in e + p collisions at HERA. We found good agreement wit te experiment. A resolution for te issue of normalization of te experimental data is suggested. In addition, te proportionality of F2 π and F p 2, wic ave caused confusion in te normalization of ZEUS data is discussed and resolved. PCAC I. INTRODUCTION ZEUS Collaboration at HERA as recently publised [] data on pion structure function, F2 π, using te leading neutron production in e + p collision. Te data do suggest tat tere is a simple relation between proton structure function, F p 2, and te pion structure function F 2 π. Tis assertion is believable and points to te direction tat tere exists a more basic and universal structure inside all adrons. In Ref.[] te normalization of σ γπ and ence, te pion structure function is fixed by two different metods: (a) dominance of one pion excange, and (b) use of te additive quark model. Te two normalizations differ by a factor of two. Te additive quark model makes no statement about te leading baryon production wile te extraction of F2 π is completely based on meson excange dynamics. A criticism of tis procedure is given in [2]. It is evident tat te issue of normalization of te data is more uncertain tan it was tougt before. ZEUS Coll. now believes tat most likely te final result will lie between te two options [3]. Te issue is wat to use for F (t) tat parameterizes te sape of te pion cloud in te proton. In tis paper we will offer an alternative, wic does not rely on tose assumptions and renders support to a normalization, wic lies between te two options used by te ZEUS Coll. In wat follows we ave used te structure function of a Constituent Quark (CQ), (in Ref. [4] it is called valon and ence te valon model) wic is universal to all adrons and from tere, wit a convolution metod, te structure function of pion is obtained. Te motivation for suc an approac is based on te fact tat suc a model for soft production as found penomenological success at s<00gev and low p T. It sould, owever, be understood tat wen s is ig enoug to generate a significant component of ard subprocesses ( s>200gev ), σ tot and average p T will bot increase and inclusive distributions will lose teir scaling beavior. Neverteless, te soft component is uncanged, and te model remains valid. Our knowledge of adronic structure is based on te adron spectroscopy and te Deep Inelastic Scattering (DIS) data. In te former picture quarks are massive particles and teir bound states describe te static properties of adrons; wile te interpretation of DIS data relies upon te QCD Lagrangian, were te adronic structure is intimately connected wit te presence of a large number of partons. It as been sown [5] tat it is possible to perturbatively dress a valence QCD Lagrangian field to all order and construct a constituent quark in conformity wit te color confinement. Terefore, te assumption of constituent quark as a valence quark wit its cloud of partons is reasonable. Te Cloud is generated by QCD dynamics. A complete description and calculational procedure for obtaining te constituent quark structure, and ence, te adronic structure functions are detailed in [6]. II. FORMALISM By definition, a valon is a universal building block for every adron. In a DIS experiment at ig enoug Q 2 it is te structure of valon tat is being probed, wile at low enoug Q 2 tis structure cannot be resolved and it beaves faras@cic.aut.ac.ir

2 as te valence quark and te adron is viewed as te bound state of its valons. For a U-type valon one can write its structure as, F2 U (z,q 2 )= 4 9 z(q u + q U U ū )+ 9 z(q + q d d U U + q s + q U U s )+... () were all te functions on te rigt-and side are te probability functions for quarks aving momentum fraction z of a U-type valon at Q 2. Tese functions are calculated in Ref. [6] in te next-to-leading order and we will not go into te details ere. Suffices to note tat te functional forms of te parton distributions in a constituent quark, (or valon), is as follows [6]: zq val. (z,q 2 )=az b ( z) c (2) CQ zq sea CQ (z,q2 )=αz β ( z) γ [ + ηz + ξz 0.5 ] (3) Te parameters a, b, c, α, etc. are functions of Q 2 and are given in te appendix of Ref.[6]. Te above parameterization of te parton distribution in a valon is for ligt quarks, u and d. For eavy quarks additional penomenological assumptions are needed to be made. It is known tat in proton te strange quark distribution is smaller tan up quark by a factor of 2 at some regions of x and by te time x reaces down to 0 4,we ave x s = xū. As for te carm quark content of proton, te picture is less clear. Early treatments of eavy parton distributions assumed tat for Q 2 >m 2 Q, te eavy quark, Q, sould be considered as massless. At te opposite extreme, te eavy quark as never been regarded as part of te nucleon sea but produced perturbatively troug poton-gluon fusion. Te number of flavors remain fixed, regardless of Q 2. Tis treatment is used in GRV 94 parton distribution [7]. Bot scemes ave teir own deficiency. For example, in te former sceme, te eavy quark sould not be treated as massless for Q 2 m 2 Q and in te latter sceme one cannot incorporate large logaritms at Q2 >> m 2 Q. CTEQ [8] on te oter and, ave used an interpolation, wic produces relative features of bot scemes. In tis treatment, te eavy quarks are essentially produced by poton-gluon fusion wen Q 2 m 2 Q and considered as massless quark wen Q 2 >> m 2 Q. In our treatment, owever, since we are dealing wit low x-values, we will assume z s = zū, andz c = αz s, were α is a factor taken to be equal to te ratio of te strange and carm quark masses wen Q 2 <m 2 carm ;and α =forq 2 >m 2 carm. Altoug suc an undertaking is not free of ambiguity, owever, tat does not cange our qualitative arguments regarding te normalization of te data. Equations (-3) completely determine te structure of a valon. Te structure function of any adron can be written as te convolution of te structure function of a valon wit te valon distribution in te adron: F 2 (x, Q2 )= CQ x dyg CQ (y)f CQ 2 ( x y,q2 ) (4) were summation runs over te number of CQ s in a particular adron. F CQ 2 (z,q 2 ) denotes te structure function of a CQ (U, D, Ū, D, etc.), as given in equation (), and G CQ (y) is te probability of finding a valon carrying momentum fraction y of te adron. It is independent of te nature of probe and its Q 2 value. Following [4], [6], and [9], for te case of pion we ave: G CQ/pion (y) = B(µ +,ν+) yµ ( y) ν (5) were G CQ is te U-valon distribution of in π + as well as te D-valon distribution in π. Similar expression for pion GCQ/pion, anti-valon distribution in a pion, is obtained by intercanging µ ν. In te above equation B(i, j) is Euler β function and G CQ (y) are non-invariant distributions satisfying te following number and momentum sum rules: 0 G CQ (y)dy = CQ 0 yg CQ (y)dy = (6) For pion, te numerical values are: µ =0.0, ν =0.06. We ave, owever, tried a range of values for µ and ν and did not find muc sensitivity on F2 π data against tis variations. In [4] [0] it is estimated tat µ = ν =0., wic is very close to te values tat are used in tis paper. Te flatness or almost flatness of te valon distribution in pion is attributed to te fact tat te valons are more massive tan te pion, so tey are tigtly bound. From SU(2) 2

3 symmetry one sould expect tat µ = ν. In our calculation, µ is sligtly different from ν, indicating a small violation of SU(2) symmetry. Tis violation is very small and te data on F2 π is not sensitive enoug to make a large difference. Significant asymmetry is observed in proton sea and its implications are discussed in Ref. [6] in te context of te valon model. Te pion as two valons (or constituent quarks), for example, π + as a U and a D, terefore, te sum in equation (4) as only two terms. Parton distributions in a pion, say π +, is obtained as: u π+ val. (x, dy Q2 )= x y G (y)u U val. ( x π + U y,q2 ) (7) d π+ val. (x, Q2 )= G D x (y) d val. π + D ( x y,q2 ) dy y (8) q sea (x, dy Q2 )= π + x y G (y)q U sea (x dy π + U y,q2 )+ x y G D (y)q sea (x π + D y,q2 ) (9) Similar relations can be written for π and π 0. In te above equations te subscripts U and D are te two valon types in π +. Equations (7, 8, 9) along wit equation (4) completes te evaluation of te pion structure function. In figures () we present F2 π(x, Q2 ), as calculated above, for te fixed Q 2 s corresponding to te ZEUS data of Ref.[]. III. DISCUSSION AND THE DATA In te previous section we outlined te procedure for calculating F2 π(x, Q2 ). Te results are sown in figures () by te square points. From te figures, it is evident tat for smaller x and lower Q 2 values, te results of te model calculations are closer to te additive quark model normalization of te ZEUS Coll. data (see figure 9 of Ref.[]). As we move towards te large x values te calculated structure function decreases and gets closer to te effective flux normalization of te data. If we are to trust in te valon model results, wic provides a very good description for te wealt of proton structure function data as well as oter adronic processes, we can conclude tat te two normalizations used by ZEUS may be relevant to different kinematical regions and lends support to te assessment made by te ZEUS collaboration tat te final results will lie between te two options[3]. It is true tat te valon model resembles te additive quark model in tat te contributions from eac valon are added up. But ere we are mainly dealing wit te parton content of eac valon, wic is derived from te perturbative QCD. Te valon distribution in pion serves only as a penomenological mimic of te pion wave function. we furter note tat in te above calculation none of te ZEUS data for pion structure function is used and no data fitting is performed. ZEUS Coll. makes te observation tat tere is a simple relationsip between te proton structure function and te effective pion flux normalization of te pion structure function (see figure 8 of Ref. []); namely F π(ef) 2 (x, Q 2 ) kf p 2 (x, Q2 ) (0) wit te proportionality constant, k =0.36. We ave calculate te rigt-and side of Eq. (0) in te valon model and compared it wit te effective flux normalization, F π(ef) 2 (x, Q 2 ), in te left and side. Te copmarison is presented in figures (). As one can see from te figures, te relationsip olds rater well at all Q 2 values. To avoid any misleadings, we empasizs tat te direct calculation of F2 π(x, Q2 ) in te valon model (Square points in figures ()) is different from F π(ef) 2 (x, Q 2 ) and ence, does not support te effective flux normalization of F2 π. In oter words, our finding merely states tat if we scale F p 2 byafactorofk =0.37 we arrive at equation (0). Figures () also indicates tat te above relationsip olds rater well at lower Q 2 values. As we move to iger Q 2 tis relationsip at te lowest x-value gets blurred, but at large x and ig Q 2 it continues to old. Since our model produces very good fit to te proton structure function data in a wide range of bot x =[0 5, ] and Q 2 =[0.45, 0000] Gev 2, we ave also attempted to investigate te relation: F π 2 (x, Q2 ) 2 3 F p 2 (2 3 x, Q2 ) () wic is based on color-dipole BFKL-Regee expansion and corresponds to te ZEUS s additive quark model normalization. We ave calculated bot sides of te equation () in te valon model. Te results are sown by te solid 3

4 lines and square points in Figures (). Altoug We get similar results as in Fig.9 of Ref.[], but tis does not say muc about te pion structure function data, because te additive quark model normalization of te data is based on te above equation. It only restates tat our model, indeed, correctly produces proton structure function. Te main ingredient of our model is te partonic content of te constituent quark wic is calculated based on QCD dynamics. Convolution of tis structure wit te constituent quark distribution in te pion appears, to some extent, to give support for te additive quark model normalization of te pion structure function data. In fact, we agree wit Ref. [3] tat te final results sould be somewere between te two normalizations used by ZEUS, being muc closer to te additive quark model sceme tan te pion flux sceme. It is wort to note tat te following relationsip also olds very well between ZEUS s pion flux normalization data and proton structure function: F π(ef) 2 (x, Q 2 )= 3 F p 2 (2 3 x, Q2 ) (2) Tis relationsip is essentially te same as Eq. (0), except for te factor 2 3 in front of x tat makes a small correction to te factor k of Eq. (0). In Figure (2) we ave presented te rigt-and sides of bot equations (0) and (2) along wit te effective flux normalization data from Ref.[] at a typical value of Q 2 =5GeV 2. Te same feature is also prevalent for te oter values of Q 2. Bot Eqs. () and (2) indicate tat, regardless of te coice of normalization sceme, te valence structure of proton, as compared to pion, is sifted to te lower x by a factor of 2 3, So tat valence x in proton corresponds to 2 3 x in pion. ZEUS as observed tat te rate of neutron production in poto-production process, in comparison to tat of pp collision, drops to alf and from tis observation ZEUS Coll. as concluded tat σ γπ 3 σ γp wereas one expects to get a rater 2 3, bot from Regge factorization and te additive quark model[2]. Tis poses a problem on te understanding of te dynamics of te interaction. A tentative resolution is tat te discrepancy can be resolved if we suppose tat in te process eac valon interacts independently. Tat is te impulse approximation. Suppose tat δ denotes te number of valons in te target proton tat suffers a collision and δ i denotes te number of collisions tat i t valon of te projectile encounters. If we define te integer σ = i δ i, ten we will ave δ σ 3δ. Furtermore, let P δ (σ) represents te probability tat out of δ independent collisions tat target valons encounter, σ valonic collisions occur. If p is te probability tat eiter of te oter valons in te projectile also interact, ten te probability for δ collisions will ave a binomial distribution aving σ δ valonic collision by i = 2 and 3 valons out of a maximum 2δ possible suc collisions []. Now, for a real poton, we can assume tat it may fluctuate into mesons wit two valons. if we denote te number of possible valonic collisions in γ p interaction by δ ten te mean number of suc collisions will be 2δ p wereas in pp collisions it will be 3δp. Te observed reduction in te rate of neutron production in two processes and te conclusion tat σ γπ 3 σ pp implies tat δ δ = 2. Tat is, tere are twice as many collisions in pp interactions tan in poto-production. IV. CONCLUSION We ave demonstrated tat te assumption of te existence of a basic structure in adrons is a reasonable model to investigate te adronic structure. Pion structure function measurement by ZEUS Coll. provides additional tests and furter validation of te Model. We ave presented a resolution to te issue of te normalization of te data. Our results are based on QCD dynamics and suggest tat te correct normalization of F2 π is closer to te additive quark model normalization for low x region. Te observed reduction in te rate of neutron production in poto-production as compared to pp collision is accounted for and concluded tat tere are twice less valonic collisions in poto-production tan in pp collision. Furtermore, it appears tat tere are simple relations among te structure functions of adrons; namely tey are proportional and te proportionality ratio seems to depend on te normalization sceme cosen. V. ACKNOWLEDGMENT I am grateful to Prof. Garry Levman for useful discussions and providing me wit te experimental data. I am also grateful to Professor Rudolp C. Hwa, wo kindly as read te manuscript and made valuable comments, and pointed out an error in Eq. (4). 4

5 [] ZEUS Collaboration, S. Cekanov et al., Nucl. Pys. B637, (2002) 3. [2] Garry Levman,Te Structure of Pion and Nucleon, and Leading Neutron Production at HERA, (2002), (submitted to Nucl. Pys B.). [3] Malcolm Derrick; private communication. [4] R. C. Hwa and M. S. Zair, Pys. Rev. D 23, 2539 (98); Rudolp C. Hwa and C. B. Yang, Pys Rev. C (2002). [5] M. Lavelle and D. McMullan, Pys. Lett. B 37, 83 (996); M. Lavelle and D. McMullan, Pys. Rep. 279, (997). [6] F. Aras and Ali. N. Korramian, ep-p/ (to appear in Pys. Rev. C). [7] M. Gluck, E. reya, anda. Vogt, z. Pys. C67 (995) 433. [8] H. L. Lai et al., Eur. Pys. J. C2 (2000) 375. [9] R. C. Hwa, Pys. Rev. D22, 593, 980. [0] R. C. Hwa and C. B. Yang, Pys. Rev. C66: (2002). [] R. C. Hwa and C. B. Yang, Pys. Rev. C65: (2002). 5